Submitted preprint.
1 Multiband Homogenization of Metamaterials in Real-Space:
2 Higher-Order Nonlocal Models and Scattering at External Surfaces
1, ∗ 2, † 1, 3, 4, ‡ 3 Kshiteej Deshmukh, Timothy Breitzman, and Kaushik Dayal 1 4 Department of Civil and Environmental Engineering, Carnegie Mellon University 2 5 Air Force Research Laboratory 3 6 Center for Nonlinear Analysis, Department of Mathematical Sciences, Carnegie Mellon University 4 7 Department of Materials Science and Engineering, Carnegie Mellon University 8 (Dated: March 3, 2021) Dynamic homogenization of periodic metamaterials typically provides the dispersion relations as the end-point. This work goes further to invert the dispersion relation and develop the approximate macroscopic homogenized equation with constant coefficients posed in space and time. The homoge- nized equation can be used to solve initial-boundary-value problems posed on arbitrary non-periodic macroscale geometries with macroscopic heterogeneity, such as bodies composed of several different metamaterials or with external boundaries. First, considering a single band, the dispersion relation is approximated in terms of rational functions, enabling the inversion to real space. The homogenized equation contains strain gradients as well as spatial derivatives of the inertial term. Considering a boundary between a metamaterial and a homogeneous material, the higher-order space derivatives lead to additional continuity conditions. The higher-order homogenized equation and the continuity conditions provide predictions of wave scattering in 1-d and 2-d that match well with the exact fine-scale solution; compared to alternative approaches, they provide a single equation that is valid over a broad range of frequencies, are easy to apply, and are much faster to compute. Next, the setting of two bands with a bandgap is considered. The homogenized equation has also higher-order time derivatives. Notably, the homogenized model provides a single equation that is valid over both bands and the bandgap. The continuity conditions for the higher-order spatio-temporal homogenized equation are applied to wave scattering at a boundary, and show good agreement with the exact fine-scale solution. Using that the order of the highest time derivative is proportional to the number of bands considered, a nonlocal-in-time structure is conjectured for the homogenized equation in the limit of infinite bands, suggesting that homogenizing over finer length and time scales is a mechanism for the emergence of macroscopic spatial and temporal nonlocality.
9 1. Introduction
10 Metamaterials have the potential to display unusual properties, and have been the focus of much attention in + + 11 mechanics and electromagnetism, e.g. [DLT 21, LR18, Sd12, HHS07, MSGP 18, DB07, KK21, BVCV17,
12 GWWM06, SRGS09, HLR14, LS18, FTP15, HE21] and numerous others. A primary focus of much of this
13 effort is to find the dispersion relations in the context of infinite periodic systems, and then tailor or optimize
14 these relations to obtain desirable properties.
15 In this paper, we go beyond computing the dispersion relation to develop homogenized models, in the
16 context of linear periodic metamaterials. The homogenized models that we develop have two key features:
17 (1) they are posed in real space and time, i.e., not in frequency space; and (2) they are posed in terms
18 of a single equation that is valid over a large frequency range and across multiple bands. These features
19 can enable the application of the models to problems posed on complex geometries and with complex
20 time-dependent behavior, rather than infinite periodic systems at steady state and at specific frequencies.
21 Our approach is based on developing approximations to the dispersion relation and then inverting it to
22 obtain the real-space model. This requires us to balance between two competing objectives: (1) an accurate
∗ [email protected] † [email protected] ‡ [email protected] 2
Figure 1. The metamaterial (left) gives a homogenized material with higher-order space-time derivatives (right), and consequent higher-order continuity conditions.
23 representation of the dispersion relation, and (2) tractability of the representation to inversion to real space
24 to obtain standard differential operators. For a single band, we find that a class of models that go back to
25 Toupin [Tou62] provides a good balance. In brief, this class of models can be considered as approximating
26 dispersion curves as rational functions; the inversion to real-space then provides standard but higher-order
27 differential operators. Specifically, the homogenized equation includes strain gradients as well as spatial
28 gradients of the inertial terms. When we extend this to higher bands, we find the appearance of higher-order
29 time derivatives as well.
30 To summarize our overall procedure, we: (1) start with a given dispersion relation that is derived
31 under assumptions of periodicity; (2) use a rational function approximation of the dispersion relation to
32 develop a homogenized equation in space and time with constant coefficients; and (3) then assume that
33 the homogenized equation is still valid even when the coefficients vary in space and the body is finite.
34 Roughly, our procedure assumes implicitly a separation of scales between the periodic microstructure and
35 the macroscopic heterogeneity.
36 To test and demonstrate the homogenized model, we turn to an important topic of current interest in the
37 community: to predict the scattering of waves at boundaries between different metamaterials or boundaries
38 between a metamaterial and a homogeneous material, e.g. [SW17, Wil20b, Wil20a, CGMP19]. We use
39 problems extracted from this body of work as examples on which to test our approach, and discuss this in
40 detail below. Our approach, in brief, is to identify the additional nonstandard continuity conditions that
41 arise at a surface of discontinuity due to the higher-order derivatives in the homogenized equations. These
42 higher-order continuity conditions then provide precisely the information required to uniquely solve for the
43 wave scattering at the boundary. A schematic view of this approach is shown in Figure1.
44 Prior Work on Scattering at Boundaries between Metamaterials and Homogeneous Materials. A
45 simplistic approach to model the scattering at a boundary might start from the usual assumption of infinite
46 periodicity to obtain the homogenized properties of the materials on either side, and then use these properties
47 in the nonperiodic setting with a boundary. However, as pointed out in [SW17], such an approach would
48 consider only the propagating waves and miss the contribution from the evanescent waves that are present
49 in the nonperiodic setting. They showed that if such homogenized models are applied to this problem,
50 the evanescent modes cannot be accounted for, and the classical continuity conditions – continuity of
51 displacement and traction – must be satisfied with propagating waves only; this leads to a violation of the
52 energy flux conservation. 3
53 [SW17] discuss an approach to deal with this problem. Their approach is to minimize the errors in
54 displacement continuity and traction continuity, subject to the conservation of energy, by considering only
55 the propagating modes. While simple and easy to use, it does not consider the microstructural information
56 in formulating this criterion, i.e., it does not account for the details of the evanescent modes. Further, it
57 can be difficult to apply this in a more general setting, for instance in a setting with complex geometry and
58 time-dependence where a decomposition into propagating modes is difficult.
59 A different “bottom-up” approach to this problem, and analogous problems, has been proposed by other
60 workers, e.g., [MMP18, CG20, CGMP19, PMM21, MM17, MM18, MPM19, GMOI19, MG18]. These
61 approaches rigorously account for the lack of periodicity, e.g., by introducing boundary-corrector functions
62 or using matched asymptotic expansions in 2-scale methods. Such methods typically require the coupled
63 solution of the cell problem – solving for the fast variable in a periodic unit cell – along with the effective
64 PDE describing the wave transmission, or the computation of the boundary-corrector functions. This
65 requirements make the approaches challenging and computationally expensive, and consequently difficult
66 to apply in a general setting. Further, these methods typically are restricted to a relatively limited frequency
67 band, making them difficult to apply to general problems with complex time-dependent loading.
68 The Proposed Approach.The approach that we propose aims to sit between the approach of [SW17] and the
69 rigorous homogenization approaches. Specifically, we aim for our homogenized models to be (1) applicable
70 to a broad range of frequencies, (2) applicable to general geometries and time-dependent loadings, (3)
71 computationally efficient and easy to compute with.
72 In the specific context of the insights provided by [SW17] on the importance of evanescent waves, our
73 homogenized models are able to capture the evanescent waves and the bandgap. Further, the higher-order
74 continuity conditions (Figure1) at the interface augment the standard continuity conditions, and provide
75 unique solutions that compare well with exact fine-scale calculations.
76 Emergence of Temporal Nonlocality with Multiple Bands. Existing dynamic homogenization techniques
77 can provide a homogenized equation for a specific band of the dispersion curve by expanding about the
78 frequency in that band, e.g. [GMOI19, CKP10, HMC16]. However, this typically does not provide a
79 single equation that covers several bands. In this work, we propose a single homogenized equation that is
80 applicable over a range of frequencies that covers multiple bands and the bandgaps. An important feature of
81 this homogenized equation is that it has higher derivatives in time, suggesting the emergence of nonlocality
82 in time due to homogenization as we increase the number of bands. We mention the seminal work by Tartar
83 [Tar89, Tar91], and others following him [Ant93], who rigorously showed the emergence of nonlocality in
84 time as an outcome of homogenization.
85 Organization. In Section2, we formulate the exact fine-scale model and discuss the specific examples that
86 we will use to compare the exact and homogenized models. In Sections3 and4, we describe the development
87 of the homogenized single-band model in 1-d and 2-d respectively, and compare its predictions with those of
88 the exact model. In Section5, we describe the development of the homogenized model for two bands with
89 a bandgap, and compare its predictions with those of the exact model. In Section6, we discuss nonlocal
90 equations as possible homogenized models that account for an infinite number of bands.
91 2. Formulation and Fine-Scale Model
92 2.A. Notation
93 We will refer to the model that considers the detailed microstructure as the fine-scale model, and its solution
94 as the fine-scale solution. Analogously, we will refer to the homogenized effective dynamical PDE as the
95 homogenized model, and its solution as the homogenized solution.
96 The primary variable, i.e. the displacement in the mechanical setting, will be denoted u(x, t), where t 97 denotes time and x is the spatial coordinate. In 1-d, x ≡ x; in 2-d, x ≡ (x1, x2) in Cartesian coordinates. 98 Our 2-d problems will be in the antiplane setting, so u is always a scalar. 4
2 ∂ ∂ 2 ∂ 99 For conciseness, we use the following representation for derivatives: ∂ ≡ , ∂ ≡ , ∂ ≡ , x ∂x t ∂t x ∂x2 2 2 ∂ ∂ 2 2 100 ∂t ≡ 2 , ∂i ≡ and so on. We use the summation convention, e.g., ∂ii ≡ ∂1 + ∂2 . ∂t ∂xi 101 We consider bodies wherein the material properties – either in the fine-scale model or in the homogenized
102 model – can be discontinuous across boundaries. Consequently, various quantities can be discontinuous at
103 these boundaries, and the jump in a quantity across a surface of discontinuity is denoted · . The surfaces
104 of discontinuity are assumed to be fixed in space. J K 105 The Bloch wavevector in the fine-scale models is denoted K = (K1,K2), and the wavevector in the 106 homogenized setting is denoted k = (k1, k2). The frequency is denoted ω, and will be normalized by the 107 midgap frequency ω0.
108 2.B. Fine-scale Governing Equations We will consider the linear wave equation in heterogeneous media. Denoting the domain of the body by Ω, we have:
2 1-d: ρ(x)∂t u(x, t) = ∂x (E(x)∂xu(x, t)) in Ω (2.1) 2 2-d antiplane: ρ(x)∂t u(x, t) = ∂i (µ(x)∂iu(x, t)) in Ω (2.2)
109 where the fine-scale material is assumed to be isotropic for antiplane shear, and µ(x) and E(x) denote
110 moduli, and ρ is the density. Let S denote a surface of discontinuity in Ω. The continuity conditions on S are given by:
1-d: u(x, t) = 0, E(x)∂xu(x, t) = 0 ∀x ∈ S (2.3) J K J K 2-d antiplane: u(x, t) = 0, µ(x)∂iu(x, t) nˆi(x) = 0 ∀x ∈ S (2.4) J K J K
111 where nˆ(x) is the unit normal to S. The continuity conditions correspond to the classical conditions of
112 displacement continuity and traction continuity respectively.
113 2.C. Problem Geometries and Boundary Conditions
(a) Single boundary geometry in 1-d, with a semi-infinite (b) Double boundary geometry in 1-d, with a laminate meta- laminate metamaterial adjoining a semi-infinite homoge- material of finite extent L (taken to be 10 unit cell widths) neous material. between two identical semi-infinite homogeneous materials.
Figure 2. Schematics of the problem geometries in 1-d.
114 a. One-dimensional Problems.In 1-d, we consider two problems, as shown in Figure2. In both problems,
115 a given traveling wave originates at x → −∞ in the homogeneous material and is incident at the boundary
116 between the homogeneous material and the metamaterial. At x → ∞, a Bloch wave form is used in the
117 semi-infinite metamaterial, and a traveling wave is used in the semi-infinite homogeneous material in the 5
118 single- and double- boundary problems respectively. This formulation, and the solution, can be found in
119 many works in the literature, e.g., Section 2 in [Wil16] and [CG20].
120 The primary quantities of interest are the magnitudes and energies of the reflected and transmitted waves
121 at steady state; we aim to compare the homogenized model predictions of these quantities with the fine-scale
122 model predictions for various frequencies.
Figure 3. Problem geometry in 2-d, with a semi-infinite laminate metamaterial adjoining a semi-infinite homogeneous material. The macroscopic boundary is normal to the laminae in the metamaterial.
123 b. Two-dimensional Problem. In 2-d, we consider the geometry shown shown in Figure3. We consider
124 plane traveling waves originating at infinity in the semi-infinite homogeneous material, and oriented at an
125 angle θ to the tangent of boundary of the metamaterial. The plane wave scatters on the boundary giving rise
126 to transmitted and reflected waves. Because the boundary is infinite and planar, the transmitted and reflected
127 waves are also plane waves and we apply a Bloch form at infinity in the semi-infinite metamaterial. This
128 formulation, and the solution, follows [Wil16, SW17].
129 The primary quantities of interest are the magnitudes and energies of the transmitted and reflected waves
130 at steady state, and we compare the homogenized model predictions and fine-scale model predictions for
131 various frequencies and various angles of incidence θ.
132 2.D. Unit Cell Material Properties
133 Throughout this paper, we consider a periodic bilayer laminate metamaterial composed of two materials a
134 and b; the geometry and notation is shown in Figure4. Table1 lists the numerical values of the material
135 properties chosen for all the examples (both 1-d and 2-d) involving a single band. Table2 lists the numerical
136 values of the material properties chosen for the examples involving 2 bands in Section5; the key difference is
Figure 4. Unit cell for laminate metamaterial. E denotes modulus and ρ denotes density. 6
137 a bigger contrast to have a larger bandgap. The bandgap frequencies used for normalization for the properties −1 −1 138 from Table1 and Table2 are ω0 = 3.448 685 rad s and ω0 = 8.646 75 rad s respectively. 139 The homogeneous material has modulus and density denoted Eh and ρh respectively. We use a numerical 140 value of ρh = 1 throughout. For the 1-d single boundary problem, we use Eh = 0.25 when we examine 141 1-band approximations and Eh = 8 when we examine 2-band approximations; for the 1-d double boundary 142 problem and the 2-d problem, we use Eh = 2. These values are chosen to provide numerical results that 143 can be easily compared when plotted.
Layer a/b Layer a/b modulus: E(a)/E(b) 1/5 modulus: E(a)/E(b) 1/20 thickness: h(a)/h(b) 4 thickness: h(a)/h(b) 1/6 density:ρ(a)/ρ(b) 1 density:ρ(a)/ρ(b) 1
Table 1. Properties of the periodic laminate con- Table 2. Properties of the periodic laminate for the sidered for all calculations (1-d and 2-d) involving a multiband calculations in Section5, with { Ea, ρ(a), single band, with {Ea, ρ(a), h(a)} = {1, 1, 4/5}. h(a)} = {1, 1, 1/7}.
144 2.E. Dispersion Relations for the Fine-scale Model
145 The dispersion relation for a periodic laminate in 1-d is classical and given in, e.g., [Wil16, Lek94]: " ! # 1 1 E(a)k(a) E(b)k(b) K = arccos cos k(b)h(b) cos k(a)h(a) − + sin k(a)h(a) sin k(b)h(b) (2.5) h 2 E(b)k(b) E(a)k(a)
q q (a) (a) (a) (b) (b) (b) (a) (b) 146 Here, K is the Bloch wavevector, k = ω/ E /ρ , k = ω/ E /ρ , h = h + h . Figure 5a
147 plots this dispersion relation using the properties in Table1.
148 The dispersion relation ω(K) for a periodic laminate in 2-d is given in, e.g., [Wil16, CG20]. It is plotted 149 in Figure 5b as a function of the wavevector components K = (K1,K2).
Re(Kh) Im(Kh) 4
3 0 / 2
1
0 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Kh (a) Dispersion relation for the laminate in 1-d with the prop- (b) Dispersion relation for the laminate in 2-d with the properties erties from Table1, restricted to the first Brillouin zone. from Table1.
Figure 5. Dispersion relations in 1-d and 2-d. 7
150 3. Single Band Homogenized Model in One Dimension
151 In this section, we examine the simple setting of a 1-d model with a single band. We begin by identifying
152 an appropriate approximation to the dispersion relation; the approximation captures the primary feature
153 that waves above a certain frequency are not supported, and at the same time is amenable to be inverted
154 to find the homogenized real-space-time dynamical equation (Section 3.A). We then identify the continuity
155 conditions that correspond to the homogenized dynamical equation in a macroscopically-heterogeneous
156 medium (Section 3.B). Finally, we apply these continuity conditions to study scattering at macroscopic
157 boundaries in specific examples (Section 3.C).
158 3.A. Approximating the Dispersion Relation We begin by comparing the dispersion relations of classical linear elasticity, strain gradient elasticity, and “microinertia” models1, in the context of homogeneous materials:
Dynamical equation Dispersion
2 2 2 2 Classical elasticity: ρ∂t u = E∂xu ρω (k) = Ek (3.1)
2 2 4 2 2 4 Strain gradient: ρ∂t u = N1∂xu − N2∂xu + . . . ρω (k) = N1k + N2k + ... (3.2)
2 2 2 4 2 2 4 ρ∂t u − ρD1∂x∂t u + ρD2∂x∂t u + ... 2 Ek + N1k + ... Microinertia: 2 4 ρω (k) = 2 4 = E∂xu − N1∂xu + ... 1 + D1k + D2k + ... (3.3)
2 159 The dispersion relations have similar behavior at k → 0. The important distinction is the behavior at 2 160 k → ∞. In this limit, classical elasticity and strain gradient models have the undesirable feature that 2 161 ω → ±∞. In contrast, microinertia models have a finite limiting value of ω if we have equal orders of
162 spatial derivatives on both sides of the momentum balance. 2 163 We notice that strain gradient models lead to ω (k) that are polynomials, and microinertia models lead to 2 164 ω (k) that are rational functions. Rational function approximations have the important advantage that they 2 165 are more flexible, e.g. they can approximate curves with finite limits at k → ∞, while also being able to
166 easily invert to real space and time to give dynamical PDE posed in terms of standard differential operators.
167 Given this advantage of rational function approximations, our strategy begins by approximating a given
168 exact dispersion relation such as (2.5) using a rational function approximation. Our approximation has the
169 form:
PN N k2i ω2 (k2) = i=1 i (3.4) approx PN 2i D0 + i=1 Dik
170 Some important features of this approximation are:
171 1. All coefficients in the rational function approximation are taken to be non-negative. This avoids
172 singularities and the frequencies are real for all k.
173 2. The exponent of the highest-order terms in the numerator and denominator are balanced to ensure a
174 finite limit at k → ∞.
175 3. The dispersion relation is an even function of k, and hence we will use only even powers of k in the
176 rational function approximation.
1 “Microinertia” refers to terms that have spatial derivatives of second-order time derivatives. Such models were introduced by Toupin [Tou62], though the terminology of “microinertia” appears to be more recent [AA06, AMPB08, AA11]. 8
177 4. While our rational function expression only approximately matches the exact given dispersion relation,
178 we constrain the approximation to ensure that we exactly capture the long-wavelength and static
179 behavior. Specifically, we require that our approximation satisfies
ω2 lim approx = c2 (3.5) k→0 k2
180 where c is the long-wavelength wave velocity. 181 5. We will denote the different rational functions by RFNN , where N refers to the exponent of the 182 highest-order term in the numerator or denominator.
183 3.B. Homogenized Dynamical Equation and Boundary continuity Conditions
184 We consider (3.4) with N = 6 for concreteness:
2 4 6 2 2 N0k + N1k + N2k ωapprox(k ) = 2 4 6 (3.6) D0 + D1k + D2k + D3k
185 Given an exact dispersion relation obtained from dynamic homogenization of a fine-scale metamaterial 186 model, we select the coefficients N0,N1,N2,D0,D1,D2 that best approximate the given exact relation.
187 We then use an elementary inverse Fourier transform to obtain the (approximate) homogenized dynamical
188 equation in real space and time:
6 4 2 6 2 4 2 2 2 2 N2∂xu − N1∂xu + N0∂xu + D3∂x∂t u − D2∂x∂t u + D1∂x∂t u − D0∂t u = 0 (3.7)
189 We emphasize that this Fourier inversion assumes implicitly that the coefficients are all constant in space.
190 We next construct the corresponding Lagrangian:
LRF66 [u](t) := Z 3 2 2 2 2 3 2 2 2 2 2 N2|∂xu| + N1|∂xu| + N0|∂xu| − D3|∂x∂tu| − D2|∂x∂tu| − D1|∂x∂tu| − D0|∂tu| dx x∈Ω (3.8)
191 and the action: Z t
SRF66 [u] := LRF66 dt (3.9) 0
192 where Ω is the body and [0, t] is the time interval of interest. The variation of SRF66 with respect to u(x, t) 193 recovers (3.7) as the Euler-Lagrange equation.
194 Here, we make the central assumption of our method. The dynamical equation (3.7) was homogeneous 195 in space, i.e. the coefficients N0,N1,N2,D0,D1,D2,D3 were constant. We now take (3.8) and (3.9) as the 196 starting points of our homogenized model, and assume that they hold even if the coefficients are functions
197 of x. Taking the variation of SRF66 with respect to u, we obtain the dynamical equation:
3 3 2 2 − ∂x(N2(x)∂xu) + ∂x(N1(x)∂xu) − ∂x(N0(x)∂xu) 3 3 2 2 2 2 2 2 (3.10) − ∂x(D3(x)∂x∂t u) + ∂x(D2(x)∂x∂t u) − ∂x(D1(x)∂x∂t u) + D0(x)∂t u = 0 9
and the corresponding continuity conditions at discontinuities:
3 3 2 qN2(x)∂xu + D3(x)∂x∂t uy = 0 (3.11) 2 q∂xuy = 0 (3.12) 3 3 2 2 2 2 q−∂x N2(x)∂xu + D3(x)∂x∂t u + N1(x)∂xu + D2(x)∂x∂t u y = 0 (3.13) ∂xu = 0 (3.14) J 2 K 3 3 2 2 2 2 2 q∂x N2(x)∂xu + D3(x)∂x∂t u − ∂x N1(x)∂xu + D2(x)∂x∂t u + N0(x)∂xu + D1(x)∂x∂t u y = 0 (3.15) u = 0 (3.16) J K
198 The macroscopic homogenized model consists of (3.10) and (3.11)-(3.16). 199 We can identify a conserved energy for the homogenized model by multiplying (3.7) by ∂tu and integrating 200 over Ω:
ERF66 [u(x)] := Z 1 3 2 2 2 2 3 2 2 2 2 2 N2|∂xu| + N1|∂xu| + N0|∂xu| + D3|∂t(∂xu)| + D2|∂t(∂xu)| + D1|∂t(∂xu)| + D0|∂tu| dx 2 x∈Ω (3.17)
201 The energy is clearly positive-definite and its time derivative can be readily computed to be 0 if there is no
202 energy input from the boundaries of Ω.
203 3.C. Numerical Comparisons between the Homogenized and Fine-scale Models
204 We consider the metamaterial with the numerical values of the properties as chosen in Section 2.D (Table1),
205 and the dispersion relation in (2.5). The exact dispersion relation and several rational function approximations
206 are shown in Figure 5a, and the values of the coefficients of the rational function approximations are given
207 in Table3. The rational function approximations are computed using a least-squares fit.
Figure 6. The exact dispersion relation and various rational function approximations.
208 a. Single-Boundary Problem.We consider the single-boundary problem described in Section 2.C (Figure
209 2a). The numerical values of the homogeneous material properties are given in Section 2.D. We solve the 10
RF22 RF44 RF66 RF88 −2 N2 0 1.224 51 × 10 0 0 −2 N3 0 0 1.350 × 10 0 −3 N4 0 0 0 1.3982 × 10 −2 −2 −4 D1 1.100 814 × 10 0 1.9503 × 10 2.745 76 × 10 −3 D2 0 4.650 93 × 10 0 0 −3 D3 0 0 2.088 92 × 10 0 −4 D4 0 0 0 2.745 76 × 10
Table 3. Coefficients in the rational function approximations obtained using least squares fits to the exact dispersion relation (2.5). Figure6 shows plots of the rational functions. N1 = 1.190 476 19 is chosen to match the classical linear elastic limit as k → 0 for all of the approximations.
210 homogenized model – (3.10) and (3.11)-(3.16) – by noticing that (3.10) has piecewise constant coefficients
211 and hence the solution is simply a piecewise superposition of exponentials. The continuity conditions are
212 then used to obtain the coefficients in the superposition solution. The solution method is easy and completely
213 standard; the difference is that we need to solve for more coefficients than in the classical case, and we also
214 have more continuity conditions to use.
215 The comparison between the predictions of the exact fine-scale model and the homogenized model is
216 shown in Figure7. Specifically, we show the predictions of normalized energy transmission across the
217 boundary as a function of frequency. The error in the homogenized model is remarkably small given the
218 significant simplifying assumptions; the error is about 5% at about 80% of the bandgap frequency.
(a) Normalized transmitted energy for the fine-scale model and (b) Normalized error in the transmitted energy between the RF66 approximation. fine-scale model and RF66 approximation.
Figure 7. Transmitted energy predictions and error as a function of frequency for the single-boundary problem in 1-d described in Section 2.C and Figure 2a.
219 b. Double-Boundary Problem. We next consider the double-boundary problem described in Section 2.C
220 (Figure 2b). The numerical values of the homogeneous material properties are given in Section 2.D. The
221 solution method for the homogenized problem closely follows the single-boundary case described above.
222 The comparison between the predictions of the exact fine-scale model and the homogenized model and
223 continuity conditions are shown in Figures8,9 and 10.
224 Figure8 shows the transmitted energy as a function of frequency. We notice the classical dips in the
225 transmitted energy that correspond to the destructive interference between the transmitted waves and the
226 multiple reflected waves. The homogenized model is able to capture the frequencies at which these dips 11
227 occur and the overall qualitative structure of the curve well even with the simplest RF22 approximation, with 228 an error of about 20% at about 80% of the bandgap frequency.
(a) Normalized transmitted energy for the fine-scale model (b) Normalized error in the transmitted energy between the and the RF22 approximation. fine-scale model and the RF22 approximation.
Figure 8. Transmitted energy predictions and error as a function of frequency for the double-boundary problem in 1-d described in Section 2.C and Figure 2b. We highlight that these predictions are based on the simplest RF22 rational function approximation that has a significant overestimate of the bandgap frequency (Figure6), yet it is able to provide good qualitative and fair quantitative predictions.
229 Figure9 shows the displacement field at a frequency about halfway to the bandgap, again using the 230 simplest RF22 approximation and comparing with the exact fine-scale model. We find a very good agreement 231 for the real and imaginary parts (i.e., the homogenized model captures the phase information correctly).
(a) Re(u), ω/ω0 = 0.3552 (b) Im(u), ω/ω0 = 0.3552
Figure 9. Comparison between the fine-scale model and the homogenized (RF22) model for ω/ω0 = 0.3552. The gray shading represents the metamaterial.
232 If we use a higher-order approximation, e.g. RF66, we are able to capture the bandgap much more 233 accurately (Figure6). Figure 10 shows an example of a wave propagating with a frequency ω/ω0 = 1.025, 234 i.e., it lies in the bandgap of the metamaterial. We notice that the evanescent wave at the boundary is captured
235 by the homogenized model, though the decay is somewhat faster than that of the exact fine-scale model.
236 4. Single Band Homogenized Model in Two Dimensions
237 We next consider the 2-d setting, and follow the overall strategy of Section3. For numerical comparisons,
238 we consider the problem described in Figure3. 12
Figure 10. Comparison of the displacement field (absolute value) between the exact fine-scale model and the homogenized RF66 approximate model. The frequency ω/ω0 = 1.025 lies in the bandgap of the metamaterial. The gray shading represents the metamaterial. The homogenized model captures the evanescent waves at the boundary of the metamaterial.
239 We begin by approximating the first band of the exact dispersion relation in Figure 5b using a rational
240 function approximation of the type:
N (0) : K ⊗ K + K ⊗ K : N (1) : K ⊗ K + ... ω2 = (4.1) approx D(0) + D(1) : K ⊗ K + K ⊗ K : D(2) : K ⊗ K + ...
(0) (1) (1) (2) 241 where N , D are second-order tensors, N , D are fourth-order tensors, and so on. All of these
242 tensors are symmetric and positive definite to avoid singularities in the approximation. 2 243 For the numerical calculations presented in this section, we keep only terms up to |K| . A least-squares
244 fit against the exact dispersion relation gives:
1.182 824 74 8.401 15 × 10−2 9.052 83 × 10−3 0 D(0) = 1, N (0) = , D(1) = sym. 1.683 05 sym. 8.483 21 × 10−3 (4.2)
245 Figure 11 compares the exact and the approximate dispersion relations.
246 4.A. Homogenized Model and Continuity Conditions
247 Transforming (4.1) to real space and time gives the 2-d homogenized dynamical equation:
(0) 2 (1) 2 (0) − D ∂t u + Dij ∂i∂j∂t u + Nij ∂i∂ju = 0 (4.3)
248 The corresponding Lagrangian L2d is: Z 1 (0) 2 (0) (1) L2d = D |∂tu| − Nij ∂ju∂iu + Dij ∂j(∂tu)∂i(∂tu) dΩ (4.4) 2 Ω Dropping the assumption of homogeneity of the coefficients and using the principle of least action, we get 13
1.6 / 0 =0.091 / =0.182 1.4 0 / 0 =0.273
/ 0 =0.364 1.2 / 0 =0.455
/ 0 =0.547 1.0 h
2 0.8 K
0.6
0.4
0.2
0.0 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 K1h
Figure 11. Contours of the exact dispersion relation (solid lines) and its rational function approximation (dashed red lines).
the following continuity conditions on the boundary,
r (0) (1) 2 z Nij ∂ju + Dij ∂j∂t u nˆ i(x) = 0 (4.5) u = 0 (4.6) J K
249 where nˆ(x) is the unit normal to the boundary.
250 4.B. Numerical Comparisons between the Homogenized and Fine-scale Models
251 We use the 2-d problem described in Section 2.C and Figure3, and with the numerical values for the material
252 properties from Section 2.D and Table1.
253 The method of solution of the homogenized equations is analogous to the 1-d case. We first notice
254 that (4.3) is linear with piecewise constant coefficients, and hence the solutions are simply a piecewise
255 superposition of exponential functions. The coefficients in the superposition are obtained by applying the
256 continuity conditions (4.5)-(4.6). This gives the scattered displacement field and scattering coefficients for
257 the homogenized model.
258 Figure 12 compares the reflection coefficients predicted by the exact fine-scale model and the homoge-
259 nized model as a function of frequency, for several frequencies and over the entire range of incident angles
260 from grazing to normal incidence. There is very good quantitative agreement over all the frequencies and
261 incident angles.
262 Figure 13 plots the displacement field of the fine-scale model and the homogenized model for a specific ◦ 263 frequency and θ = 30 . Visually, the agreement is very good.
264 Our consideration of this problem is motivated by [SW17] that examined the same problem in a different
265 way. They highlighted that when the metamaterial lamination is normal to the metamaterial boundary, 14
(a) ω/ω0 = 0.1624 (b) ω/ω0 = 0.3625
(c) ω/ω0 = 0.397 (d) ω/ω0 = 0.57
Figure 12. Comparison between the reflection coefficients predicted by the fine-scale and homogenized models as a function of incident angle for several frequencies.
(a) (b)
Figure 13. Real part of the displacement fields in 2-d from the RF22 homogenized model (a) and the fine scale model ◦ (b). The boundary is along x2 = 0, the wave is incident at 60 to the normal, and ω/ω0 = 0.3624. Visually, the agreement is very good. 15
266 there are a finite number of propagating waves and an infinite number of evanescent waves generated in the
267 metamaterial. Standard dynamic homogenization can account only for the propagating waves because it
268 uses the setting of an infinite periodic medium. If such homogenized models are applied to this problem,
269 the evanescent modes cannot be accounted for, and the classical continuity conditions – continuity of
270 displacement and traction – must be satisfied with propagating waves only; this leads to a violation of
271 the energy flux conservation. In the case of multiple propagating waves in the 2-direction, the classical
272 continuity conditions are not sufficient to determine the transmission and reflection coefficients.
273 The proposed approach of using higher-order homogenized models gives additional continuity conditions
274 at the metamaterial boundary. Further, the higher-order homogenized model supports evanescent waves.
275 These features address the issues raised in [SW17] and enable good quantitative predictions.
276 5. A Single Homogenized Equation to Describe Two Bands with a Bandgap
277 We now consider the question of modeling multiple bands, and focus on two bands with a bandgap to
278 illustrate the idea in a simple setting. Prior work, e.g., [CKP10], has derived approximations for higher
279 bands, but these approaches are restricted to a single band in the vicinity of an a priori selected frequency.
280 In contrast to such approaches, we obtain a single homogenized equation that is seamlessly valid across the
281 entire frequency range spanning both bands, without having to specify a priori the band or limit the frequency
282 range of interest. Such a model is essential to enable the modeling of complex time-dependent transient
283 loadings and waves that are composed of a broad range of frequencies. For instance, shock loading involves
284 a very broad range of frequencies, and it is essential to be able to use a single model across frequencies to
285 describe such a situation.
Consider two bands, and denote their rational function approximations by ωˆa(k) and ωˆb(k). The simplest representation has the form:
nk2 ω2 = =: (ˆω (k))2 (5.1) 1 + dk2 a pk2 ω2 = ω2 − =: (ˆω (k))2 (5.2) b 1 + qk2 b
286 where n, d, p, q, ωb are constants.
287 To obtain a composite dispersion relation that can represent both bands together, we simply use the
288 product of the individual dispersion relations:
2 2 2 2 (ω − ωˆa(k) )(ω − ωˆb(k) ) = 0 (5.3)
289 The resulting composite dispersion relation is easy to invert to real space and time to obtain the multiband
290 homogenized equation:
4 4 2 4 4 2 2 4 2 2 2 2 ∂t u − (q + d)∂t ∂xu + dq∂t ∂xu − (−ωb dq + pd − nq)∂t ∂xu + (p − ωb q − ωb d − n)∂t ∂xu 2 2 2 2 2 4 (5.4) + ωb ∂t u − ωb n∂xu + (ωb nq − np)∂xu = 0
2 2 2 291 To simplify notation, we define: A1 := q + d, A2 := qd, A3 := −ωb dq + pd − nq, A4 := p − ωb q − ωb d − 2 2 2 292 n, A5 := ωb ,A6 := ωb n, A7 := ωb nq − np.
293 To find the Lagrangian LMB[u](t), we first multiply (5.4) by a test function v(x, t) and integrate over 294 the spatial domain Ω and an arbitrary time interval [t1, t2]. Using standard integration-by-parts operations, 16
295 we can simplify to write:
Z Z t2 2 2 2 2 2 2 2 2 2 2 ∂t u · ∂t v + A1∂t ∂xu · ∂x∂t v + A2∂t ∂xu · ∂x∂t v + A3∂t∂xu · ∂x∂tv x∈Ω t1 (5.5) 2 2 + A4∂t∂xu · ∂x∂tv − A5∂tu · ∂tv + A6∂xu · ∂xv + A7∂xu · ∂xv dx dt = 0
296 where we have ignored boundary terms in space and time, by assuming that the initial conditions are given 297 and hence v and its time derivatives are zero at t = t1. 298 This can be used to write the action:
Z t2 SMB = LMB dt (5.6) t1
299 where the Lagrangian is:
LMB := Z 2 2 2 2 2 2 2 2 2 2 2 2 2 |∂t u| + A1|∂t ∂xu| + A2|∂t ∂xu| + A3|∂t∂xu| + A4|∂t∂xu| − A5|∂tu|r + A6|∂xu| + A7|∂xu| dx x∈Ω (5.7)
300 Extremizing SMB without assuming that the coefficients are constant gives the homogenized equation:
4 4 4 2 2 2 2 2 2 2 2 2 ∂t u−∂t ∂x (A1∂xu)+∂t ∂x A2∂xu −∂t ∂x A3∂xu +∂t ∂x (A4∂xu)+A5∂t u−∂x (A6∂xu)+∂x A7∂xu = 0 (5.8) and the continuity conditions:
4 2 2 4 2 2 2 2 qA1∂t ∂xu − A4∂t ∂xu + A5∂t u + A6∂xu − ∂t ∂x A2∂xu + ∂t ∂x A3∂xu − ∂x A7∂xu y = 0
(5.9) u = 0 (5.10) J K 4 2 2 2 2 qA2∂t ∂xu − A3∂t ∂xu + A7∂xuy = 0 (5.11) ∂xu = 0 (5.12) J K
301 In AppendixA, we use LMB and SMB to obtain the conserved energy (A5) and consequently an 302 expression for the energy flux across a surface (A6). The energy flux expression is required to compute the
303 transmitted and reflected energies.
304 Remark 5.1. An important feature of our homogenized model is the presence of higher-order derivatives in
305 time. For a generic higher-order-in-time Lagrangian, Ostrogradsky’s theorem [Ost50] shows that the corre-
306 sponding Hamiltonian (energy) has linear instabilities, referred to as Ostrogradsky’s instability. However,
307 our homogenized model is marginally stable in the sense of Lyapunov when the coefficients are chosen in
308 the appropriate range.
309 5.A. Numerical Comparisons between the Homogenized and Fine-scale Models
310 We use the 1-d single-boundary problem described in Section 2.C and Figure 2a, with the numerical values
311 for properties listed in Section 2.D (Table2).
312 An important aspect of the homogenized model is that the metamaterial is described using higher-order
313 time derivatives. To have a consistent mathematical description, we therefore require that the homogeneous
314 material also be modeled using higher-order time derivatives of the same order. However, we have consid-
315 erable freedom in our choice of how we do this. For simplicity, we consider the homogeneous material as a 17
316 bilayer laminate metamaterial with identical layers. To get the correct higher-order time derivatives, we use
317 two bands to construct the higher-order model of the homogeneous material. Four different choices of the
318 second band are shown in the dispersion diagram in Figure 14.
Figure 14. Dispersion relations for the metamaterial (Kh > 0) and the homogeneous material (Kh < 0). The periodicity of the artificial uniform laminate that represents the homogeneous material is chosen to be the same as the periodicity of the laminate metamaterial. The first band for the homogeneous material is the linear band (shown in red), and 4 different choices of approximation of the second band are shown by the curves with symbols. The dispersion bands for the metamaterial are shown by the black curves (exact) and red curves (rational function approximation).
319 We solve the homogenized model by first using that (5.8) is linear with piecewise constant coefficients,
320 and hence the solutions are the standard piecewise superposition of exponentials. The coefficients within
321 the superposition are then obtained by applying the continuity conditions (5.9)-(5.12).
322 Comparisons of the transmitted energy for the 4 choices are shown in Figure 15. The bandgap frequencies
323 can be identified as the range of frequencies over which there is no transmission, and is captured very well
324 by all choices. The difference in the 4 choices is in the modeling of the second band of the homogeneous
325 medium. The first band is considered as the straight line and is exactly captured by the rational functions.
326 However, the linearity of the second band can be captured only approximately by the the second-order
327 rational functions. The results in Figure 15 show that the predictions improve as the maximum frequency of
328 the approximation approaches the maximum frequency of the second band in the metamaterial.
329 Further, for the fourth choice, we highlight that frequencies in the second band can correspond to
330 wavevectors in either of the two bands; in other words, there are two possible wavevectors for a given
331 frequency when the frequency is sufficiently high. Figure 15d compares using an incoming wave with
332 wavevector chosen from the extended first band (“Type 1”) and chosen from the second band (“Type 2”).
333 The latter is the physically-appropriate choice and we find that it also performs much better, giving very
334 good qualitative and fair quantitative agreement.
335 6. Nonlocal Limit Models
336 Section5 showed the emergence of 4th-order time derivatives when we considered 2 bands. In principle, it
337 is straightforward to consider as many bands as one wishes by continuing the approach in (5.3). For instance, 18
(a) Case 1 (b) Case 2
(c) Case 3 (d) Case 4
Figure 15. Comparison of the transmitted energies predicted by the exact fine-scale model (solid blue curves) and the homogenized model (dotted blue curve) for the 4 choices of approximation shown in Figure 14. In each case, the band gap is captured very well. Case 4 above has two possible approaches, and the second approach shows very good agreement.
338 if we wish to consider N bands, we can write each band in terms of a rational function approximation ωˆi(k), 339 and construct the composite dispersion relation as:
N Y 2 2 ω − ωˆi (k) = 0 (6.1) i=1
2N 340 The resulting dispersion relation will include (even) powers of ω of up to order ω , and the corresponding
341 homogenized model in real time and space will include (even) time-derivatives of up to order 2N.
342 We recall that the fine-scale model will generically have an infinite number of bands. This raises
343 the question of what limit homogenized model we might expect if we aim to represent all bands in our
344 homogenized model. We could think of each higher-order time derivative as containing information from
345 further out in time; the (heuristic) limit of an infinite order time-derivative might then correspond to a
346 nonlocal operator in time.
347 Our heuristic reasoning parallels work in peridynamics [Sil00], wherein formal arguments based on
348 Taylor expansions have been used to relate nonlocal integral operators in space to an infinite series of spatial
349 derivatives of increasing order, following the idea of matching dispersion relations [WA05, Day17, BD18,
350 SPGL09, SDP16]. From that perspective, one could think of peridynamics as a special case of the limit
351 models discussed here, in that it is the limit of a single band model.
352 In [Day17], it was shown formally that the limit of microinertia models – corresponding to a single-band
353 model in the perspective put forward in this paper – is consistent with a nonlocal integral equation of the 19
354 form: Z Z 2 0 2 2 0 0 0 0 0 ρ∂t u(x, t) + C1(x, x ) ∂t u(x, t) − ∂t u(x , t) dx = C2(x, x ) u(x , t) − u(x, t) dx x0∈Ω x0∈Ω (6.2) 0 0 355 where C1(x, x ) and C2(x, x ) are spatial kernels.
356 A possible simple limit model that has nonlocality in both space and time is of the form:
(u(x, t − T0) − 2u(x, t) + u(x, t + T0)) Z Z τ=T0 0 0 0 0 0 + C1(x, x )K(τ) u(x, t − τ) − 2u(x, t) + u(x, t + τ) − u(x , t − τ) + 2u(x , t) − u(x , t + τ) dτ dx x0∈Ω τ=0 Z 0 0 0 = C2(x, x ) u(x , t) − u(x, t) dx x0∈Ω (6.3)
357 K(τ) is a time-kernel, and T0 is the fixed extent of non-locality in time. The structure of the term that is 358 nonlocal in both space and time follows from (6.2), and the term that is nonlocal in time alone is put forward
359 because we found – by trial and error – that it has an appropriate bandstructure. As an example, choosing e−y2 e−y2 360 T = 0.8,K(τ) = 1, C (y) = 0.01 √ , C (y) = 2 √ gives the dispersion relation: 0 1 π 2 π
2 2 −k 2 −k sin ω e 4 sin ω 25 sin (0.4ω) 99e 4 99 − − − + = 0 (6.4) 50ω 50ω 8 50 50
361 Figure 16 shows the dispersion relation with an infinite number of dispersion bands.
Figure 16. Dispersion relation of the space-time non-local model from (6.4).
362 We note in closing that the continuity conditions for nonlocal operators are much weaker than in models
363 with higher-order derivatives.
364 7. Discussion
365 We have presented an approach to develop homogenized models for metamaterials that are posed in space and
366 time, and consequently potentially applicable to general settings such as complex geometries and complex
367 time-dependent loading and to a broad range of frequencies. The homogenized models have higher-order 20
368 derivatives in space and time, leading to nonstandard continuity conditions at the boundaries between
369 metamaterials.
370 While the predictions of the homogenized models compares well with exact fine-scale models, there
371 are a number of important open questions. The most central question is a justification of the method in
372 a more “bottom-up” fashion. While our approach provides models that are much simpler to use than the
373 rigorous approaches, e.g. [CG20], it is important to understand the connection between our approach and
374 these rigorous approaches; in particular, if it is possible to view our approach as a simplified version of the
375 rigorous approaches under certain assumptions.
376 A second open question is how to perform practical numerical calculations with our model, given the
377 nonclassical higher-order derivatives. In the context of finite elements (FE) methods, two options appear
378 feasible. First, second-order rational functions provided fairly good accuracy and have spatial derivatives of
379 4th-order. Hence, they are amenable to Isogeometric FE methods which provides sufficient continuity for
380 such derivatives [KB19]. Second, mixed FE methods are a possible approach for problems that require very
381 smooth interpolations, though mixed methods can require care to formulate correctly. The implementation
382 of numerical methods will enable application of the approach to numerous problems of current interest in
383 the broad area of metamaterials.
384 We highlight that this work, and the many others cited here, are all in the linear regime. Homogenization
385 of dynamics in the nonlinear regime is largely unexplored, though we mention a study of wave scattering at
386 an interface between a phase-transforming solid and a linear elastic solid [AK92]. On the other hand, we
387 also note that the linear dispersion behavior is important even in nonlinear problems, for instance to govern
388 the effective dissipation [DB07].
389 We note that some aspects of our overall approach – namely, the approximation of dispersion relations
390 and subsequent inversion to real space and time – is broadly similar to strategies used in the construction of
391 nonreflecting boundary conditions (NRBC) [BT80, GK90]. In the NRBC literature, such an approach has
392 been used to construct wave equations that propagate waves in a tailored manner.
393 In the context of linear homogenization, we note other interesting works that obtain homogenized models
394 that have features in common with our homogenized models. Specifically, the elimination of fine-scale
395 degrees of freedom has been found to give rise to effective nonlocality in space by [PSP18, DW96], and also
396 to give rise to very nonstandard dynamical behavior in [MW07, PZ12]. Further, [Tar91, Ant93, Tar89] find
397 the emergence of memory effects – i.e, nonlocality in time – due to homogenization.
398 Our work provides heuristic insight into possible physical mechanisms that can drive the emergence of
399 spatial and temporal nonlocality. Specifically, it provides the perspective that perhaps peridynamics is the
400 limit of a spatially averaged model that consider only a single dispersion band; the physical origin of the
401 nonlocality in space is the coarse-graining of the fine structure. Analogously, a richer model that considers
402 all dispersion bands will provide the full spatio-temporally nonlocal model that is equivalent to averaging
403 in time and space. However, it is not clear such a limit nonlocal model will have practical advantages over
404 models with a few higher-order derivatives: the former are much more expensive to compute, and there are
405 no compelling applications. We can contrast this to the case of peridynamics that was developed to model
406 dynamic fracture.
407 Acknowledgments
408 We thank George Gazonas, Robert Lipton, and Luc Tartar for useful discussions; NSF (1635407),
409 ARO (W911NF-17-1-0084, MURI W911NF-19-1-0245), ONR (N00014-18-1-2528), and AFOSR (MURI
410 FA9550-18-1-0095) for financial support; and Pittsburgh Supercomputing Center for computing resources.
411 A. Energy Flux for the Nonlocal-in-Time Model
412 Given a Lagrangian with higher-order time derivatives, an expression for the corresponding Hamiltonian
413 can be obtained by considering the time derivative of the Lagrangian. We first show this for a single particle 21
414 system with an equation of motion given by:
4 2 ∂t up + C∂t up − up = 0, (A1)
415 where up ≡ up(t) describes the particle motion and C is a given constant. The corresponding Lagrangian 416 for which (A1) is the Euler-Lagrange equation is: 1 L [u ](t) = |∂2u |2 − C|∂ u |2 − |u |2 (A2) p p 2 t p t p p
417 Taking the time derivative of (A2), we get: dL p =∂2u · ∂3u − C∂ u · ∂2u − u · ∂ u dt t p t p t p t p p t p 4 2 d 2 2 2 3 (A3) =∂tup · ∂t up + C∂t up − up + |∂t up| − C|∂tup| − ∂tup · ∂t up | {z } dt =0 from (A1)
418 Therefore, we have the conserved energy: 1 1 1 E [u (t)] := |∂2u |2 −C|∂ u |2 −∂ u ·∂3u −L = |∂2u |2 − C|∂ u |2 −∂ u ·∂3u + |u |2 (A4) p p t p t p t p t p p 2 t p 2 t p t p t p 2 p
419 We apply a similar approach to the Lagrangian in (5.7). We take the time derivative of (5.7), and use the
420 chain rule, integration-by-parts, and (5.8) to obtain terms over the bulk and over the boundary. The bulk
421 terms give the conserved energy:
EMB[u(x, t)] := Z 1 2 2 A5 2 3 A1 2 2 A4 2 3 2 |∂t u| − |∂tu| − ∂tu · ∂t u + |∂t (∂xu)| + |∂t(∂xu)| − A1∂t(∂xu) · ∂t (∂xu) x∈Ω 2 2 2 2 ! A A A A + 2 |∂2(∂2u)|2 + 3 |∂ (∂2u)|2 − A ∂ (∂2u) · ∂3(∂2u) − 6 |∂ u|2 − 7 |∂2u|2 dx 2 t x 2 t x 2 t x t x 2 x 2 x (A5)
422 and the boundary terms give an expression for the flux through a surface:
4 2 4 2 4 2 2 EMBflux [u(x, t)] :=2 ∂tu · A1∂x∂t u − ∂tu · ∂x(A2∂x∂t u) + ∂x∂tu · A2∂x∂t u + ∂tu · ∂x(A3∂x∂t u) 2 2 2 2 2 −∂x∂tu · A3∂x∂t u − ∂tu · A4∂x∂t u + ∂tu · A6∂xu + ∂x∂tu · A7∂xu − ∂tu · ∂x(A7∂xu) (A6)
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